Theorem 1.1 [1]}
Let \(M\)  be an \(n\) -dimensional complete Riemannian manifold. For any \(p > \frac{n}{2}\) , \(H \in \mathbb {R}\)  (assume \(r \le \frac{\pi }{2\sqrt{H}}\)  when \(H > 0\) ),
\(||\varphi ||_{2p}(r) \le \left( \frac{(n-1)(2p-1)}{2p-n} ||Ric^{\,H}_{-}||_{p}(r) \right)^{\frac{1}{2}}.\) 
